Using the weighted space method, introduced by P. Găvruţa and L. Găvruţa [Int. J. Nonlinear Anal. Appl. 1, No. 2, 11–18 (2010; Zbl 1281.39038)], the authors prove the generalized Hyers-Ulam stability of the single variable functional equation \(y(x)=F(x,y(x),y(\eta(x)))\) and apply it to extend some results of L. Cădariu et al. [An. Univ. Vest Timiş., Ser. Mat.-Inform. 47, No. 3, 21–26 (2009; Zbl 1240.39058)] and J. A. Baker [Proc. Am. Math. Soc. 112, No. 3, 729–732 (1991; Zbl 0735.39004)]. As a consequence they prove the stability of the linear equation \(y(x)=g(x)y(\eta(x))+h(x)\). They also discuss the generalized Hyers-Ulam stability for a general class of the nonlinear Volterra integral equations in Banach spaces.